Radical Theory for Granded Rings

Abstract

In this paper we propose a general setting in which to study the radical theory of group graded rings. If 31 is a radical class of associative rings we consider two associated radical classes of graded rings which are denoted by 3l G and 3l Ki . We show that if 31 is special (respectively, normal), then both 31 and 3l ni are graded special (respectively, graded normal). Also, we discuss a graded version of the ADS theorem and the termination of the Kurosh lower graded radical construction.

Graded varieties of graded rings

“…Let F be a finite field in 1)g(P, N). Then F satisfies (1) and (2). By Lemma 2.3, F is a subfield of a field of order pm for some p E P and some m E N(p).…”

On the restricted graded Jacobson radical of rings of Morita Context

“…This means that the ring M 2 (J) is a Z 4 −graded ring which proves that M 2 (J) ∈ J Z4 . In Proof It is clearly shown from Proposition 2 in [2] that J G is a graded supernilpotent radical class since the Jacobson radical J is a supernilpotent radical class. P Furthermore, Morita context has an important role on radical theory of rings.…”

“…In general radical class, we can define the restricted graded radical class γ G = {A is a G−graded ring |A u ∈ γ} of γ to the category of G−graded rings. In general radical classes, some properties of the γ G were described in [2]. We call J G as G−graded Jacobson radical class.…”

Graded radical graded semisimple classes

“…He shows that if C is a proper subclass of all associative rings, then C is a radical semisimple class if and only if there is a strongly hereditary finite set C(~) of finite fields such that R E C if and only if R is isomorphic to a subdirect sum of fields in C(~-) or equivalently R E C if and only if every finitely generated subring of R is isomorphic to a finite direct sum of fields in C(.T). In [4] Fang and Stewart give some examples of graded radical graded semisimple classes and mention that it remains an open question how to characterize such classes. We answer their question in this paper.…”

“…By a graded homomorphism f of degree (h, k) between two graded rings R and S we mean a ring homomorphism f: R ~ S such that f(Rg) C Shgk for all g E G and h, k E G. A graded isomorphism is denoted by ~, and a complete direct sum (direct product) by :~-~*. The symbols Z, Z+, ISI respectively denote the set of integers, the set of positive integers, and the cardinality of the set S. For most of the undefined terms in graded rings we refer to [5] and for those in radical theory for graded rings we refer to [4]. 0236-5294/95/$4.00 (~ 1995 Akad@miai Kiad6, Budapest H. YAHYA By [x], where x E R, we denote the subring of a graded ring R generated by x.…”